| L(s) = 1 | − 3·2-s + 6·4-s + 3·5-s + 3·7-s − 10·8-s − 9·10-s − 3·11-s − 13-s − 9·14-s + 15·16-s + 7·17-s + 3·19-s + 18·20-s + 9·22-s + 3·23-s + 6·25-s + 3·26-s + 18·28-s + 4·29-s − 5·31-s − 21·32-s − 21·34-s + 9·35-s − 2·37-s − 9·38-s − 30·40-s − 41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 1.34·5-s + 1.13·7-s − 3.53·8-s − 2.84·10-s − 0.904·11-s − 0.277·13-s − 2.40·14-s + 15/4·16-s + 1.69·17-s + 0.688·19-s + 4.02·20-s + 1.91·22-s + 0.625·23-s + 6/5·25-s + 0.588·26-s + 3.40·28-s + 0.742·29-s − 0.898·31-s − 3.71·32-s − 3.60·34-s + 1.52·35-s − 0.328·37-s − 1.45·38-s − 4.74·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.380061082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.380061082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 3 T + 22 T^{3} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T - 6 T^{2} - 78 T^{3} - 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 24 T^{2} + 8 T^{3} + 24 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 58 T^{2} - 220 T^{3} + 58 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 36 T^{2} - 50 T^{3} + 36 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 55 T^{2} - 256 T^{3} + 55 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 86 T^{2} + 302 T^{3} + 86 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 71 T^{2} + 116 T^{3} + 71 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 64 T^{2} - 104 T^{3} + 64 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 - 14 T + 145 T^{2} - 1028 T^{3} + 145 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 205 T^{2} + 1508 T^{3} + 205 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - T + 26 T^{2} + 404 T^{3} + 26 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $D_{6}$ | \( 1 - 8 T + 57 T^{2} - 688 T^{3} + 57 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 11 T + 244 T^{2} + 1586 T^{3} + 244 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 8 T + 179 T^{2} + 920 T^{3} + 179 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T - 3 T^{2} - 520 T^{3} - 3 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 229 T^{2} + 1232 T^{3} + 229 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 219 T^{2} + 2052 T^{3} + 219 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 33 T + 570 T^{2} + 6568 T^{3} + 570 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.375757539500270428583267452795, −7.64505031620058481563203189489, −7.56456030324053744079270194696, −7.53161551185861690236774491543, −7.23960164687173150727319771864, −6.90897266533509030199053305931, −6.87976357111305687994787129829, −6.04544570229625922977554260925, −5.92895976119256469435791416814, −5.83773592149012265885980781220, −5.52327010564622602133984201579, −5.29149124876157713310563113285, −5.13544407037646207865830470036, −4.51284679004621206678298880350, −4.11139061853621326477343850955, −4.05464090548458572120320279292, −3.25554100253115218425793341542, −2.90532806023335387889832660096, −2.80595442905507045445129546824, −2.38399835308151187172869194962, −2.15204519825300887911007438662, −1.62531890434199560998945431872, −1.29526058308459081231307607058, −0.946701776765772005968830684738, −0.60212299356919442244351812269,
0.60212299356919442244351812269, 0.946701776765772005968830684738, 1.29526058308459081231307607058, 1.62531890434199560998945431872, 2.15204519825300887911007438662, 2.38399835308151187172869194962, 2.80595442905507045445129546824, 2.90532806023335387889832660096, 3.25554100253115218425793341542, 4.05464090548458572120320279292, 4.11139061853621326477343850955, 4.51284679004621206678298880350, 5.13544407037646207865830470036, 5.29149124876157713310563113285, 5.52327010564622602133984201579, 5.83773592149012265885980781220, 5.92895976119256469435791416814, 6.04544570229625922977554260925, 6.87976357111305687994787129829, 6.90897266533509030199053305931, 7.23960164687173150727319771864, 7.53161551185861690236774491543, 7.56456030324053744079270194696, 7.64505031620058481563203189489, 8.375757539500270428583267452795
